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Description

Main issue of High Resolution Doppler Imagery is related to robust statistical estimation of Toeplitz Hermitian positive definite covariance matrices of sensor data time series (e.g. in Doppler Echography, in Underwater acoustic, in Electromagnetic Radar, in Pulsed Lidar). We consider this problem jointly in the framework of Riemannian symmetric spaces and the framework of Information Geometry. Both approaches lead to the same metric, that has been initially considered in other mathematical domains (study of Bruhat-Tits complete metric Space & Upper-half Siegel Space in Symplectic Geometry). Based on Frechet-Karcher barycenter definition & geodesics in Bruhat-Tits space, we address problem of N Covariance matrices Mean estimation. Our main contribution lies in the development of this theory for Complex Autoregressive models (maximum entropy solution of Doppler Spectral Analysis).

Specific Blocks structure of the Toeplitz Hermitian covariance matrix is used to define an iterative & parallel algorithm for Siegel metric computation. Based on Affine Information Geometry theory, we introduce for Complex Autoregressive Model, Kohler metric on reflection coefficients based on Kohler potential function given by Doppler signal Entropy. The metric is closely related to Kohler-Einstein manifold and complex Monge-Ampere Equation. Finally, we study geodesics in space of Kohler potentials and action of Calabi & Kohler-Ricci Geometric Flows for this Complex Autoregressive Metric.